Evaluation of Design Methods for Prestressed Concrete Members with Stirrups Using a New Traditional Shear Database

Structures 23 (2020) 621–634

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Evaluation of Design Methods for Prestressed Concrete Members with Stirrups Using a New Traditional Shear Database Mallikarjun Perumalla, Arghadeep Laskar

T

⁎

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Prestressed Concrete Shear Database Shear Design Traditional Shear Failure Web-Shear Flexure-Shear

Prestressed concrete (PC) is increasingly used nowadays for the construction of structures subjected to higher loads. Shear design of PC members is complicated and ambiguous as there is no rational and universally accepted shear design procedure. A comprehensive shear database of PC members helps researchers to understand the complicated and ambiguous shear design procedure. In the present study code provisions on shear design based on 45-degree truss analogy (IS 1343:2012 and ACI 318-14), Modified Compression Field Theory (AASHTO LRFD Design Specifications), Variable angle theory (EC-2) have been investigated along with newly proposed shear strength equations. A shear database covering the traditional shear failures of PC beams with stirrups has been developed to assess the shear strength and behaviour of PC beams. Out of 683 shear tests available from the literature on PC beams a total of 274 PC beams with stirrups has been investigated to assess the degree of conservativeness and accuracy of shear design code provisions and shear strength equations. Investigations on the shear code provisions with various variables that govern the shear strength show that serious attention should be given to the prediction of the accurate failure modes for the web-shear and flexure-shear critical specimens. The oversimplifications in the new shear strength equations have also been highlighted in the present study.

1. Introduction Design procedures should be simple, as well as easy to understand and implement in practice. If empirical equations are used in design procedures, then the understanding of the design procedures becomes complicated. Some shear design procedures for prestressed concrete (PC) members are simple but do not include all the parameters involved in various shear transfer mechanisms. Other shear design procedures are complicated and are used by practising engineers through step-bystep methods without much understanding of their background. This has led to the development of new shear design equations for PC members by various researchers. The two primary modes of traditional shear failure in PC members are web-shear and flexure-shear failure. Cracks develop in concrete when the principal tensile stress in the member reaches the cracking strength of concrete. The cracks are oriented perpendicular to the direction of the principal tensile stress. The principal tensile stresses are parallel to the longitudinal axis of PC members subjected to pure tension or pure flexure. Hence, the cracks in these members are oriented perpendicular to the longitudinal axis of the members. Web-shear cracks, inclined to the member axis, are formed near the centroid of the concrete sections where the shear stress

⁎

is predominant as shown in Fig. 1(a). Web-shear cracks are also called diagonal cracks. The inclined cracking shear can be calculated by equating the principal tensile stress at the cross-sectional centroid of the member to the tensile strength of concrete. Flexural-shear cracks originate vertically similar to flexural cracks and thereafter become inclined as they propagate to the mid-depth of the section. Typical flexure-shear cracks are shown in Fig. 1(b). Empirical equations are used to predict the flexure-shear cracking force as it cannot be predicted from the principal stresses at the mid-depths of sections. It is essential to have two different shear strength equations for estimating the webshear and flexure-shear strengths since the failure mechanisms associated with these two failure modes are different. The web-shear and the flexure-shear strengths can also be represented through a single equation through proper implementation of the shear span-to-depth ratio (a/d) that distinguishes between the two types of failure. Thus, the traditional shear failure of PC members can be distinctly classified as web-shear failure (for members with a/d ratio less than 2.5) or flexureshear failure (for members with a/d ratio greater than 2.5) based on the recommendations of Nawy [1]. Ritter [2] idealised the reinforced concrete member as a truss where longitudinal reinforcement and stirrups are assumed to act as bottom

Corresponding author. E-mail address: [email protected] (A. Laskar).

https://doi.org/10.1016/j.istruc.2019.10.021 Received 12 June 2019; Received in revised form 5 October 2019; Accepted 28 October 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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M. Perumalla and A. Laskar

σcp ρl ρvfy ζ a/d Asw, Asv, bw bv bv,eff

Nomenclature VACI 318 Calculated Shear Strength as per ACI 318-14 Method VAASHTO Calculated Shear Strength as per AASHTO LRFD shear strength provisions VIS 1343 Calculated Shear Strength as per IS 1343:2012 shear strength equations VEC-2 Calculated Shear Strength as per EC-2 code provision VUH Calculated Shear Strength as UH Method VCladera Calculated shear strength as per Cladera’s Method Vexp Experimental Shear Strength Vn, max Maximum Shear Strength Vcw Web-Shear Strength as per ACI 318-14 Method Vci Web-Shear Strength as per ACI 318-14 Method Vco Uncracked shear strength as per IS 1343:2012 Vcr Cracked shear strength as per IS 1343:2012. Vp Vertical component of shear strength due to prestress f’c Cylinder strength of concrete λ Material strength factor according to ACI 318-14 method. θ Strut inclination angle β Concrete shear transfer factor as per AASHTO LRFD Method τ Design shear stress in concrete calculated as per IS 1343:2012 γc Partial safety factor for concrete

c dv D d, dt z ft fcp fp fy, fyt Kp Mo s, st, sv

Compressive stress due to prestress Longitudinal reinforcement ratio Shear Reinforcement Index size effect parameter Shear span-to-depth ratio Av Total cross-sectional area of stirrups Width of the web Width of web adjusted in the presence of ducts Effective width of the web calculated as per Cladera's Method. Neutral axis depth calculated as per Cladera's Method Effective shear depth calculated as per AASTHO LRFD Method Overall depth of the section Effective depth of the beam Depth of lever arm calculated as per EC-2 code provision Tensile strength of concrete Compressive stress due to prestress Characteristic tensile strength of the strand Characteristic tensile strength of the stirrup steel. Prestressing factor parameter according to Cladera's Method Moment necessary to produce zero stress according to IS 1343:2012 Spacing of stirrups

Fig. 1. Shear Failure Modes in Prestressed Concrete Members.

and vertical chords, respectively. Concrete elements in between the cracks are assumed to act as diagonal struts carrying compressive stresses. Later in 1922, Morsch [3] extended the discrete truss model approach by Ritter to a continuous truss model approach. Both Ritter and Morsch assumed a 45-degree inclination for the compression strut by neglecting the tensile stresses of cracked concrete. In 1978, the Compression Field Theory (CFT) was developed by Collins and Mitchell [4] by applying the “Tension Field Theory” in concrete members to calculate the inclination angle of the compression struts. The Modified Compression Field Theory (MCFT) [5] was introduced in 1986 to account for the tensile stress in cracked concrete, which was assumed to be zero in CFT. Truss models, CFT and MCFT were unable to accurately predict the concrete contribution to the shear strength of a concrete member. Thus a large number of experimental investigations have been conducted to formulate empirical expressions for predicting the concrete shear strength in a member [6]. Various shear design provisions use different approaches to estimate the shear strength of a concrete member. This has led to the development of simpler and more accurate shear design equations for PC by several researchers. Creation of a comprehensive shear database is essential to understand the complex behaviour of PC beams and to investigate the merits and demerits of individual shear strength equations included in various codes and proposed by other researchers.

for PC members has been studied through the development of a traditional shear database for PC members with shear reinforcement. The new comprehensive traditional shear database consisting of 274 PC beams with rectangular, T, I and U-shaped sections exclusively with shear reinforcement tested under shear failure adds the latest shear tests on PC members to the previous shear databases developed by various researchers for reinforced concrete (RC) and PC members [7–10]. The specimens included in the database have failed in a traditional shear failure mode like web-shear and flexure-shear. The shear databases developed so far by various researchers on PC specimens have very few specimens with shear reinforcement that failed in either web-shear or flexure-shear failure modes. None of them compared the degree of accuracy and conservativeness of different code provisions with newly developed shear strength equations for traditional shear failure PC specimens. The ratio of experimental to analytical shear strengths of the 274 PC beams has been obtained in the present study using various shear design provisions and shear strength equations proposed by other researchers. The variation of shear strength ratios of web-shear and flexure-shear failure specimens with various parameters that govern the shear failure behaviour has been studied. Comparative analysis of the strength ratios highlights the drawbacks of the current shear design provisions and modifications required for further development of simpler shear design equations.

2. Research Significance

3. Shear Design Provisions and New Shear Strength Equations

In the present study, the degree of the conservativeness of various shear design code provisions and shear strength equations developed

The various shear design provisions and shear strength equations for PC members included in the present study for analysing the 274 PC 622

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values of the inclination angle of the compression strut (θ) and concrete shear transfer factor (β) [5] are obtained from design Tables B5.2-1 and B5.2-2 included in AASHTO LRFD Bridge Design Specifications [12] and the specified upper limit of the nominal shear strength is 0.225fcbvdv. The strengths of concrete and reinforcing steel are limited to 68.95 MPa (10 ksi) and 517.12 MPa (75 ksi) respectively.

beams in the developed shear database are ACI 318-14 [11], AASHTO LRFD 2010 [12], IS 1343:2012 [13], EC-2 [14], Cladera’s Method [15] and UH Method [16]. A brief description of each of the shear strength calculation procedure is provided in Sections 3.1 through 3.6. Notations used in Sections 3.1 through 3.6 have been taken from the respective design guidelines or publications by the respective researchers. 3.1. ACI 318-14

3.3. IS 1343:2012 ACI 318-14 code provisions are based on load and resistance factored design approach with a strength reduction factor of 0.75 for shear design. The detailed method specified in ACI 318-14 for the shear design of PC member has been considered in the present study as it covers the effects of more number of design parameters (especially the compressive prestress, fpc) on the shear strength of PC members than the approximate simplified method. The ultimate shear strength of prestressed concrete members is taken as the contribution of concrete and steel in ACI 318-14 design specification. A 45-degree truss is used for calculating the steel contribution. The concrete contribution is taken as the smaller of web-shear (Vcw) and flexure-shear cracking strength (Vci) as shown in Table 1. The concrete strength √fc is restricted to 8.3 MPa in the calculation of Vci and Vcw [11] due to the lack of test data on high strength concrete. The yield strength of stirrup is also restricted to 415 MPa in the steel strength estimation to ensure the ductile failure of PC members [11]. It should be noted that the vertical component of prestressing force is not included for calculating the flexural shear strength of PC members. A maximum shear capacity, as shown in Table 1, is specified to ensure that transverse reinforcement yields prior to the crushing of concrete. The web-shear cracking strength is derived as the shear force causing a principal tensile stress of 0.33λ√fc (ACI 318-14, 359-360). A tensile strength of 0.5λ√fc is used to calculate the cracking Moment Mcr at flexural cracking (ACI 318-14, 359).

The shear resistance at a particular section of a PC member consists of concrete and steel contribution as per IS 1343:2012 design specifications. Shear design in IS 1343:2012 is based on the limit state of collapse with a load factor of 1.5 and partial material safety factors of 1.5 for concrete and 1.15 for steel. 45-degree truss model is adopted for calculating the steel contribution [13]. The concrete contribution is taken as the smaller of uncracked strength Vco (similar to web-shear strength in ACI 318-14) and cracked strength Vcr (same as flexure-shear strength in ACI 318-14) as shown in Table 1. The fck value in the Vn, max equation shown in Table 1 is restricted upto 55 MPa. An intrinsic partial material factor of safety (FOS) of 1.5 is included in calculating the value of ft as 0.24√fck for estimating the uncracked strength. Ib/Q is also conservatively approximated as 0.67bD (applicable for a rectangular cross-section) in the estimation of Vco. An additional FOS is used in the Vco term to account for the unforeseen variation in fcp. The yield strength of stirrups is restricted to 415 MPa. The expression for Vcr term is also formulated based on a rectangular cross-section. A FOS of 0.8 is used with the design stress in concrete due to prestress (fcp) to calculate the value of Mo required for the estimation of Vcr.

3.4. Euro Code-2 EC-2 [14] design specifications use Truss analogy proposed by Ritter [2] for calculating the shear resistance of members with shear reinforcement. The shear design in EC-2 code provisions is carried out at the ultimate limit state. EC-2 has a limitation of 90 MPa on the concrete strength and restricts the yield strength of shear reinforcement between 400 MPa and 600 MPa. The shear load on PC members with web reinforcement is assumed to be resisted only by the shear reinforcement in the EC-2 shear design specifications as shown in Table 1. The inclination of the truss angle is in between cot−1 1 and cot−12.5. Maximum shear capacity of the shear reinforcement is limited by considering the crushing strength of the concrete compression strut. Partial safety factors [14] for material are used in calculating the shear resistance of PC members with shear reinforcement.

3.2. AASHTO LRFD The shear design procedure of AASHTO LRFD Bridge Design specification [12] follows a sectional method based on MCFT [5]. AASHTO code provisions are also based on load and resistance factor design approach where the nominal shear strengths are calculated with a strength reduction factor of 0.9. The shear strength of a prestressed concrete member in the sectional method is the sum of concrete and steel contributions and the vertical component of prestressing force. Concrete contribution to the shear strength, as shown in Table 1, is obtained from the tensile stresses across the inclined shear cracks [5]. Steel contribution, as shown in Table 1, is a function of the variable inclination angle of the compression strut (θ) instead of the 45-degree inclination of the compression strut assumed in ACI 318-14 [11]. The Table 1 Different shear code and shear strength equations. Shear code/Strength equation

Concrete Contribution

ACI 318-14 [11]

Vcw = (0.29λ fc′ + 0.3fpc ) b w dp + Vp

Vci = 0.05λ fc′ b w dp + Vd + AASHTO LRFD [12] IS 1343:2012 [13]

Cladera’s Method [15]

UH Method [16]

Maximum Shear Strength Equation

Av f yt d

(Vc + 0.66 fc′ b w d)

s

Vi Mcre Mmax

0.0316β fc′ b v dv

Av f y dv cot θ

Vco = 0.67bD ft2 + 0.8fcp ft

0.87f y Asv

Vcr = ⎛1 − 0.55 ⎝ – ⎜

EC-2 [14]

Steel Contribution

f pe fp

′

⎞ ς bd + c ⎠ ⎟

dt sv

(

0.63 fck b w d

V Mo M Asw z f ywd s

⎡ f 2/3 c ς Kp ⎢0.30 c + 0.5 1 + d γc ⎢ ⎣ 1.17 (a / d)0.7

0.25fc′ b v dv

s

b bw

)

Vs ⎤ ⎥ b v, eff bd

d

cot θ

(ds − c ) cot θ

Asw f st ywd

αcw b w zν1

fcd (cot θ + tan θ)

αcw b w ds ν1 fc′

⎥ ⎦

fc′ b w d

Av f y

623

(

d s

)

−1

1.33 fc′ b w d

cot θ 1 + cot2 θ

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The shear span-to-depth ratio is considered to estimate the size effect in the members. Cladera’s Method adopts the maximum shear strength of EC-2 design specifications [14] for checking the maximum strut contribution.

3.5. Cladera’s Method A new method called Cladera’s Method was proposed to estimate the shear strength of concrete members based on a shear-flexural mechanical model developed by Mari et al. [15]. Cladera’s Method is also based on the ultimate limit state method. According to this method, the shear strength of a PC member is taken as the sum of concrete, steel and vertical component of the prestressing force. The concrete and steel contributions proposed in this method are shown in Table 1. Concrete contribution accounts for the shear strength due to uncracked concrete, dowel action and cracked web contribution. The tensile strength of concrete used to estimate the concrete contribution is limited to 4.6 MPa. The modulus of elasticity of concrete is limited to 39 GPa. The angle of compression strut or crack inclination is taken as the function of neutral axis depth c/ds and c/dp for RC and PC members respectively.

3.6. UH Method A simple and reasonably accurate equation for predicting the shear strength of PC was developed at the University of Houston [16]. Shear strength calculations in the UH Method are performed as per the ultimate limit state method with a strength reduction factor of 0.9. The proposed equation is a function of shear span-to-depth ratio, concrete strength and web area. The UH Method unifies web-shear strength with the flexure-shear strength of PC members by directly considering the ‘a/ d’ term in its formulations. Loov’s shear resistance concept [17] was

Table 2 Details of Specimens and SSR calculations. Reference

No.

a/d

fc (MPa)

D (mm)

L (m)

Vexp/VACI

Sandra et al. [18] Garber et al. [19] De Wilder et al. [20]

2 3 2

92 to 118 81.4 78

711 1168 630

4.70 12.19 5.00

Long et al. [21] Ross et al. [22] Nabipay and Dagmar [23] Labib et al. [24] Hovell et al. [25] Wang and Ding [26] Lee et al. [27] Laskar et al. [16] Gustavo et al. [28] Avendano et al. [29]

3 2 2 10 3 1 7 5 2 4

73 to 76 73 48 48 to 117 83 to 91 42 45 to 85 64.5 to 74.5 38.8 78 to 95

600 1575 270 686 to 711 1594 500 1200 713 1321 914

Choulli et al. [30] Heckman [31]

4 18

81 to 96 66 to 88

Runzell et al. [32]

2

3.68 3.00 3.13 to 3.91 3.2 2 3.5 1.77 to 3 2.6 2.2 2.5 1.6 to 4.3 4.2 2.92 to 3.75 3.1 2.18 to 2.20 2.7

70

Sudhira De Silva et al. [33] Sudhira De Silva et al. [34] Byung and Kim [35] Ma et al. [36] Shahawy and Batchelor [37]

1 4 2 4 8

Tawfiq [38] Rangan [39]

1 12

Alshegeir and Ramirez [40]

3

Hartman et al. [41] Xuan et al. [42] Maruyama et al. [43] Kaufman and Ramirez [44] Durrani and Robertson [45] Elzanaty et al. [46] Lynberg [47] Moayer and Regan [48] Bennett and Debaiky [49] Cederwall et al. [50] Bennett and Balasooriya [51] Hanson and Hulsbos [52]

4 5 7 1 6 16 6 5 22 10 13 3

Hanson and Hulsbos [53] Bruce [54]

9 17

MacGregor [55]

17

Mattock and Kaar [56] Hernandez [57]

13 15

Total

274

3 3 3.66 1.13 & 1.23 2.17 to 3.14 2.54 2.51 to 2.63 4.32 to 4.70 3 2.93 2.93 2.2 2.75 3.80 & 5.80 2.78 3.5 to 3.7 3 1.7 to 4.2 1 to 4 2.54 to 3.39 1.9 to 4.43 2.82 to 4.09 2.69 to 3.63 1 to 4.50 2.79 to 4.73 SSR Average

Vexp/ VAASHTO

Vexp/ VIS1343

Vexp/VEC-2

Vexp/VUH

Vexp/VCladera

1.69 1.32 1.43

1.97 2.09 2.89

2.14 2.93 2.04

2.00 1.84 3.06

1.69 1.65 1.79

1.21 1.76 0.99

4.90 13.00 1.77 5.18 to 6.92 8.8 to 9.3 3.50 10.00 7.32 17.20 8.54

1.18 0.99 3.06 1.44 1.35 1.98 2.34 1.36 0.98 1.71

1.90 1.02 3.32 1.02 1.50 1.84 2.44 2.64 1.41 2.88

1.67 2.73 3.69 3.45 2.66 2.56 2.97 1.96 1.13 2.79

1.96 1.27 3.27 1.73 1.26 2.49 2.73 3.35 1.46 3.35

1.23 1.16 2.85 1.33 1.20 0.95 1.33 1.17 1.55 2.22

1.32 1.09 1.88 1.52 1.63 1.53 2.04 1.21 0.63 2.09

750 1016

6.1 to 8.8 15.70

1.47 1.60

2.14 4.04

2.12 2.39

1.86 6.41

1.62 1.77

1.58 1.83

11.50

1.26

1.94

1.49

2.51

1.25

0.99

100 43 to 49 43 & 62 57 to 74 48

1372 to 1600 500 500 1200 1290 1117

3.00 3.00 10.00 18.6 8.24 to 12.2

5.47 4.54 1.56 1.68 1.23

5.74 4.88 1.77 1.76 2.03

5.74 3.93 1.84 2.96 1.64

10.41 9.02 1.53 1.60 1.28

2.92 2.79 1.44 1.41 1.11

3.26 3.26 1.61 1.27 1.33

70 29 to 45

1118 615

12.20 3.75

1.41 1.54

2.08 1.46

1.87 2.53

2.11 2.41

1.19 1.36

0.81 2.11

61 to 62

712 & 915

3.05 & 3.66

1.19

2.21

1.64

2.47

1.13

1.26

74 to 78 31.5 to 38 38 to 45 58 39 to 46 40 to 74 28 to 34 38 to 44 33 to 58 25 to 57 34 to 45 45 to 47

515 & 546 490 490 711 508 355 & 457 600 320 330 260 254 to 457 457

3.96 3.00 3.00 4.88 3.35 3.81 5 2.00 3.66 3.00 2.74 to 3.43 2.54 & 3.39

1.60 2.13 1.17 1.35 1.44 1.30 1.32 1.44 1.26 1.70 1.48 1.41

1.44 2.62 1.30 1.96 2.85 2.01 1.80 1.85 2.42 1.85 1.82 3.53

3.47 2.69 1.51 1.68 1.88 1.71 2.13 1.82 1.73 2.05 3.29 1.84

2.37 2.74 1.54 3.20 4.01 2.15 1.79 2.38 2.55 2.38 2.73 3.55

1.55 1.46 0.76 1.23 1.21 1.88 1.33 1.20 1.49 0.93 1.48 1.76

1.43 1.78 0.98 1.33 1.39 1.16 1.59 1.16 1.23 1.15 2.17 1.20

42 to 51 19 to 27

457 305 & 355

2.79 to 5.33 2.75

1.22 1.19

2.59 1.73

1.62 1.57

3.69 1.57

1.29 1.28

1.11 1.33

17 to 32.1

305

2.75

1.20

2.33

1.50

2.71

1.41

1.13

42 to 47 17 to 32

647 305 & 355

10.07 2.75

1.32 1.36

1.97 2.31

1.96 1.51

2.02 2.32

1.19 1.36

1.07 1.06

1.49

2.24

2.14

2.74

1.44

1.44

624

318

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(Vtest/Vcal) has been obtained for all the 274 beams in the developed shear database to compare the degree of conservativeness of different shear design provisions and shear strength equations. The ideal value of shear strength ratio (SSR) is a unity that is the shear strength at failure should match with the predicted value. SSR less than unity is undesirable as it indicates over-estimation of shear strength. Thus the SSR values from an ideal estimate are expected to be close to unity with a lower coefficient of variance and a minimum number of strength overpredictions (corresponding to SSR values less than unity). No strength reduction factors have been used to calculate Vcal for different shear provisions. However, the intrinsic FOS discussed in Sections 3.1 through 3.4 have been considered in calculating the SSRs. The Vcal values have been calculated at the point of application of the external loads during the testing of the individual specimens to maintain uniformity in the calculations of Vcal values obtained from the various shear strength equations. It has also been observed that most of the shear failures (web-shear as well as flexure-shear) of PC members subjected to concentrated loads, occur near the loading points. Table 2 provides the SSR calculation of various shear design provisions and shear strength equations discussed in Section 3. Statistical parameters like mean, standard deviation (SD) and coefficient of variation (COV) of the SSR values for all the shear strength equations discussed in Section 3 have been obtained along with histograms, box plots and Vexp values comparison with respective code provision and shear strength equations have also been obtained. Statistics of computed SSR of four shear design provisions (EC-2, IS 1343:2012, ACI 318-14, AASHTO LRFD) and two shear strength equations (Cladera’s Method and UH Method) for all the 274 specimens in the developed database are shown in Table 3. It can be observed from Table 3 that all the shear strength estimates have a mean SSR greater than unity. It implies that all the code provisions and the shear strength equations included in the present study

used in calculating the shear capacity of PC members in this method. Concrete contribution, shown in Table 1, is estimated from the shear stress along the failure plane, unlike ACI 318-14 and AASHTO LRFD shear design provisions. Contribution of steel, shown in Table 1, is based on the minimum shear resistance concept [17] with the number of stirrups crossing the crack as (d/s −1). 4. Traditional Shear Failure Specimens Database A shear database of 274 PC beam specimens that failed in shear (traditional shear failure like web-shear and flexure-shear) has been developed in the present study to evaluate the various shear design provisions and shear strength equations. The overall depth of the specimens varies from 254 mm to 1600 mm. Table 2 provides the necessary details of the specimens considered in the shear database. The additional details of the two hundred and seventy four PC specimens included in the shear database are given in Appendix-A. 4.1. Filtering Criteria Very few databases are available for a traditional shear failure of PC beams exclusively with shear reinforcement. A rigorous literature review has been carried out to obtain the PC beams with stirrups that have failed in either web-shear or flexure-shear failure mode. A total of 683 shear tests on PC beams has been studied to identify the traditional shear failure specimens. The filtering criteria used to select the 274 specimens for the developed database is shown in Fig. 2. 4.2. Shear Strength Ratio Calculation The ratio of shear strength of the specimens at the failure to the shear strength calculated using a specified shear design procedure

Fig. 2. Filtering Criteria for Traditional Shear Database. 625

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Table 3 Shear Strength Ratio Statistics. No of Specimens

Parameters

ACI 318-14

AASHTO LRFD

IS 1343:2012

EC-2

UH Method

Cladera’s Method

274 (133)

Mean SD COV Maximum Minimum Unconservative Cases

1.49 (1.61) 0.55 (0.69) 0.37 (0.43) 5.46 (5.46) 0.89 (0.91) 11 (5)

2.24 (2.33) 0.87 (1.08) 0.39 (0.46) 5.74 (5.74) 0.71 (0.71) 7 (7)

2.14 (2.43) 0.75 (0.76) 0.35 (0.27) 5.74 (5.74) 1.08 (1.08) 0 (0)

2.74 (3.05) 1.64 (2.12) 0.60 (0.69) 12.22 (12.22) 0.83 (0.83) 2 (2)

1.44 (1.45) 0.39 (0.40) 0.27 (0.27) 2.96 (2.96) 0.63 (0.88) 21 (7)

1.44 (1.60) 0.49 (0.53) 0.34 (0.33) 3.58 (3.37) 0.51 (0.51) 41 (13)

UH Method are less scattered (i.e. histogram distribution curve is less spread). Box plots provide the distribution of SSR values and also filter out SSR values with significant deviation from the mean SSR values obtained using any of the code provisions and shear strength equations. Closer boxes without any outlier or with outliers closer to the maximum value indicate a better prediction of shear strengths. The box plots for code provisions and shear strength equations is shown in Fig. 3b. EC-2 and IS 1343:2012 code provisions have outliers significantly higher than the maximum observed values observed in the corresponding histogram curve. AASHTO LRFD and ACI 318-14 code provisions have mean towards the median line, and the outlier SSR values are closer to the highest observed value. UH Method and Cladera’s Method have very close mean, 25 percentile and 75 percentile SSR values which suggest the accuracy of these two methods. However, compared to Cladera’s Method, outlier values obtained from the UH Method are closer to the highest obtained SSR values from the corresponding methods. Thus SSR values obtained using the UH Method are more accurate and more consistent than other code provisions and Cladera’s Method.

provide a conservative estimation of shear strength. Strength predictions from Cladera’s Method and UH Method (based on Loov’s concept) matched better with the test results with mean SSR values of 1.446 (COV 0.39) and 1.439 (COV 0.49) respectively for the 274 specimens. However, Cladera’s Method over-estimated the strength of more number of specimens than the UH Method. The overall prediction of shear strength using the EC-2 method (based on variable angle truss concept) deviated the most from the test results with the mean SSR value of 2.74 (COV 0.60). The SSR values obtained using the EC-2 shear design provisions have the highest COV (0.60) indicating inaccurate representation of the relevant parameters. SSR values from ACI 318-14 shear design provisions matched better with the experimental results with mean SSR values of 1.49 (COV 0.55). The SSR values of 133 large beams (with depth greater than 500 mm) were separately studied to obtain the degree of conservativeness of the various shear strength equations for practical cases. The statistical SSR data for the large beams are also shown in Table 3 within parenthesis. It can be seen that the shear strength predictions from the UH Method are slightly better than Cladera’s Method for the large beams. It can also be observed that the mean of SSR values obtained from all the estimates, except the UH Method, increases for large beams. This indicates that all the shear strength equations (except UH) are based on test results of smaller specimens. It can thus be concluded that UH Method (based on Loov’s concept) is most suitable for practical shear strength prediction of large beams. The histogram and box plots (25 percentile, median, 75 percentile) of the SSR distribution and its range for various code provisions and shear strength equations have been shown in Fig. 3. The variation of Vexp and Vcal for various code provisions and the shear strengths equations is also shown in Fig. 3. It can be observed that the values of Vcal increase with the increase in Vexp values for all code provisions and shear strength equations used in the present study. However, the increment in the Vcal values with increasing values of Vexp is not identical for various code provisions and shear strength equations. The shear strength predictions from all shear strength equations and code provisions, except IS 1343:2012 match well with test results at lower Vexp values. It can be observed that UH Method predicted the increment in shear strength even at higher Vexp values. IS 1343:2012 and ACI 318-14 which are based on the 45-degree truss analogy are unable to predict the shear strength of the specimens at higher experimental values. This is mainly because these code provisions have been developed based on old shear tests on small specimens and need to be updated with the latest test results. However, IS 1343:2012 deviated a lot at the higher experimental values than compared with ACI 318-14. Even though EC-2 predicted the increment in shear strength at higher Vexp values but the SSR histogram distribution shows the higher conservativeness of the SSR values. The distribution of the SSR values obtained using EC-2 shear strength provisions is significantly away from unity. Among all the code provisions ACI 318-14 has the least spread in the distribution curve of SSR histogram compared to IS 1343:2012, AASHTO LRFD, EC2. Most of the SSR values obtained using the UH Method are in the lower range compared to the SSR values obtained using Cladera’s Method. Cladera’s Method has less vertical distribution closer to one than UH Method. Thus, UH Method has accurately predicted the shear strength at higher Vexp values, and the SSR values obtained using the

4.3. Demerit Points Classification (DPC) The accuracy of strength prediction of the various shear design provisions and shear strength equations has also been verified by using the Demerit Points Classification (DPC) proposed by Collins [58]. DPC has been used by several researchers for investigating their newly proposed shear strength equations for concrete structures [59–60]. According to Collins [58], the behaviour of concrete structures can be distinguished into seven categories ranging from extremely dangerous to extremely conservative depending on the SSR values (as shown in Table 4). The efficiency of a particular method or shear strength equation is determined from a total score obtained by adding the product of the demerit point assigned to each category with the percentage number of specimens having SSR values corresponding to that category. Thus a lower score indicates a better prediction of experimental strength values. A total of 274 specimens available in the shear database have been considered in the present study. The values of total demerit point scores for the 133 large beams (overall depth greater than 500 mm) available in the shear database are also shown in Table 4 within parenthesis. Shear strengths of none of the 274 specimens predicted from all the design equations and shear strength equations result in SSR values corresponding to the extremely dangerous category. Among all the shear design equations and shear strength equations, EC-2 has the highest DPC score, and ACI 318-14 has the least DPC Score. UH Method has the lower DPC score among the two shear strength equations. The SSR values obtained from all the code provisions are mostly within the ranges of SSR values corresponding to the approximately safety and extremely conservative categories. Most of the SSR values obtained using UH Method and Cladera’s Method correspond to the approximately safety and conservative categories. Shear strength predictions from these two methods result in few SSR values corresponding to the dangerous category. However, shear strength prediction of large beams by UH Method does not result in SSR values corresponding to the low safety or extremely dangerous 626

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Fig. 3. (a) Statistical Comparison of Design Methods. (b) Box Plots of SSR values from various Shear Strength Equations.

5. Investigation of Shear Design Parameters

categories. The total DPC scores obtained from shear design equations (AASHTO LRFD, IS 1343:2012 and EC-2) and Cladera’s Method are almost twice the DPC score from the UH Method for large beams. UH Method has the lower DPC score compared to ACI 318-14 for large beams. This infers that the shear strength predictions from UH Method are more consistent and matched better with the experimental failure values in the practical cases with large beams.

The calculated SSR values have been used to study the effect of various design parameters on the shear strength of PC members. The various parameters investigated in the present study include shear span-to-depth ratio (a/d), concrete strength (f’c), prestressing force (σcp), longitudinal reinforcement ratio (ρl) and shear reinforcement 627

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Fig. 3. (continued)

provisions indirectly include the a/d ratio for estimating the concrete contribution for flexure-shear. It can be clearly observed that the shear strength predictions from ACI 318-14 and IS 1343:2012 match more with the experimental results for beams with a/d ratios greater than 3.5. This could be due to the exclusion of the effect of a/d ratio in the estimation of the concrete contribution to web-shear strength of PC beams in the ACI 318-14 and IS 1343:2012 shear strength equations. Shear strength predictions from Cladera’s Method and UH Method matches well with the experimental shear strengths. However, Cladera’s Method overestimates the shear strength of some of the specimens with a/d values ranging from 2.5 to 5.0, in the Kani’s valley region [62]. On the other hand, UH Method overestimates the shear strength of the specimens with a/d ratios ranging from 1.5 to 3. The variation of SSR values with a/d ratios for AASHTO LRFD and EC-2 design provisions, which do not consider the a/d ratios in the estimation of shear strengths, indicates undesirable (either unconservative or over-conservative) strength predictions at a/d ratios ranging from 1.0 to 3.5 as shown in Fig. 5(b). Shear strength predictions for some specimens using the EC-2 shear design provisions are very conservative because the design equations not only ignore the concrete contribution for the specimens with stirrups but also do not include shear span-todepth ratio in shear strength calculations. ACI 318-14 and IS 1343:2012 also underestimate the shear strength of specimens with a/d ratios ranging from 1.0 to 3.5 due to the exclusion of the a/d term in the webshear strength calculations. Thus, shear strength estimated from both code provisions have a similar trend of variation with respect to shear span-to-depth ratio (a/d). This indicates that it is essential to directly include the shear span-to-depth ratio in the shear equations for predicting the web-shear as well as flexure-shear strengths.

index (ρvfy). Maximum shear strength equations are used to ensure ductility and prevent concrete crushing failures in PC members with higher shear reinforcement index values (i.e. with higher steel stirrup ratios). The SSR vs shear reinforcement index plots in Fig. 4 distinguishes the specimens in which the shear strength is governed by the corresponding maximum shear strength equation of the various code provisions and shear strength estimation methods included in the present study. IS 1343:2012 and Cladera’s Method has the highest number of specimens in which the shear strength is governed by the maximum shear strength equation of the respective method. The shear strength of very few specimens was governed by the maximum shear strength equations in the AASHTO LRFD and UH Methods. Most of the over-reinforced PC specimens have the shear reinforcement index more than 10. It can be observed that if the shear reinforcement index is greater than 10, then most of the code provisions and shear strength equations are governed by the corresponding maximum shear strength equation. Thus a total of 16 (out of 274) specimens with very high reinforcement index (greater than 10) have been excluded to study the influence of various design parameters on traditional shear failure (Flexure-shear and web-shear) of PC members since these specimens have failed prior to reaching the full shear capacity of the concrete and the steel. Dermit point classification (DPC) categories of SSR values have also been used to study the influence of the various shear parameters in Sections 5.1 through 5.5. 5.1. Shear Span-to-Depth Ratio Shear failure of PC members depends on the shear span-to-depth ratio. It was shown that there is a change of 225 percent in shear stress when the a/d ratio changed from 2.35 to 1.17 [61]. Hence a/d ratio is an important parameter which is responsible for the arch action [62]. Some shear provisions like IS 1343:2012 and ACI 318-14 indirectly include the a/d ratio for estimating the concrete contribution to flexureshear but not to web-shear strengths. UH Method includes the effect of a/d ratio for predicting web-shear as well as flexure-shear strengths of PC members. The variation of SSR values with a/d ratios for the various code provisions and shear strength equations is shown in Fig. 5. The SSR values obtained using IS 1343:2012, ACI 318-14 & Cladera’s Method have been plotted together in Fig. 5(a) as these design

5.2. Concrete Strength Strength of concrete is an important parameter that governs the shear strength of PC members. UH Method and AASHTO LRFD shear design provisions directly use the concrete strength (√fc) to estimate the concrete contribution and maximum shear strength in PC members. IS 1343:2012 and ACI 318-14 uses concrete strength (√fc) for estimating the concrete contribution indirectly in terms of tensile strength and directly for estimating maximum shear strength equation. The variation

Table 4 Evaluation of Shear Strength Equations by DPC. Category

SSR

Demerit Point

ACI 318-14

AASHTO LRFD

IS 1343:2012

EC-2

UH Method

Cladera’s Method

Extremely Dangerous Dangerous Low Safety Approximate Safety Conservative Extremely Conservative Total Score

less than 0.5 0.5–0.65 0.65–0.85 0.85–1.30 1.30–2.00 greater than 2

10 5 2 0 1 2

0 (0) 0 (0) 0 (0) 39* (26) 54 (65) 7 (9) 68# (83)

0 (0) 0 (0) 1 (2) 9 (13) 38 (33) 53 (52) 145 (141)

0 (0) 0 (0) 0 (0) 4 (2) 52 (31) 45 (67) 141 (165)

0 (0) 0 (0) 0 (1) 4 (5) 37 (38) 59 (56) 155 (151)

0 (0) 0 (0) 3 (0) 38 (44) 52 (49) 7 (7) 74 (62)

0 (0) 2 (3) 4 (5) 41 (20) 42 (56) 11 (17) 82 (113)

*SSR Percentage Value i.e., (107/274 = 39%) ; # 68 =(10 × 0)+(5 × 0)+(2 × 0)+(0 × 39)+(1 × 54)+(2 × 7) 628

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Fig. 4. Variation of SSR Values Governed by Conventional and Maximum Shear Strength Equations.

(FS) and web-shear (WS) have been distinguished in the SSR plots in Fig. 6. The SSR values calculated using for IS 1343:2012, EC-2 and AASHTO LRFD increases with concrete strength in case of both FS and WS critical specimens. This increment is more prominent in WS critical specimens than in FS specimens. However, the shear strengths of some specimens have been significantly underestimated using the shear strength equations of AASHTO LRFD and EC-2 code provisions. Shear

of SSR values obtained from the UH Method, ACI 318-14 and IS 1343:2012 with the concrete strength of the specimens have been plotted together in Fig. 6(a). The variation of SSR values with concrete strength for Cladera’s Method, which uses the tensile strength of concrete (fc2/3) to estimate the concrete contribution, has been plotted in Fig. 6 (c). EC-2 uses the concrete strength to estimate the maximum shear strength of PC members. Specimens which failed in flexure-shear

Fig. 5. Variation of SSR values with Shear Span-to-Depth Ratio. 629

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Fig. 6. Variation of SSR Values with Concrete Strength.

include the sectional prestress (fpc) for calculating the flexure-shear strength. The shear strength predictions from IS 1343:2012 (Fig. 7(a)) match better with the experimental results at lower values of prestressing force even though the shear design equations directly include the sectional prestress for calculating WS as well as FS strengths. There is a large scatter of SSR values from the other methods, as shown in Fig. 7(b). Extremely conservative predictions of shear strength can be observed from EC-2 and AASHTO LRFD code provisions at lower compressive stress due to prestress. Cladera’s Method overestimates the strength of FS critical specimens at lower compressive stresses and underestimates the strength of WS critical specimens at higher compressive stresses as observed in Fig. 7(b). The variation of SSR values with effective prestress (ratio of effective prestress to the ultimate stress in the tendon) is shown in Fig. 8. SSR values obtained using ACI 318-14 and IS 1343:2012, which directly use the effective prestress in shear strength calculation have been plotted together in Fig. 8 (a). SSR values obtained using AASHTO LRFD design provisions and Cladera’s Method, which indirectly use the effective prestress to calculate the concrete contribution have been plotted in Fig. 8(b). SSR values obtained using UH Method and EC-2 design provisions, which do not use the effective prestress to calculate the shear strength of PC members are plotted together in Fig. 8(c). It can be observed that the SSR values from most of the methods increase as the effective prestress increases up to 55–60 percent and then decrease. This can be observed predominately for shear strengths estimated using EC-2 (based on variable angle truss analogy) and AASHTO LRFD (based on MCFT) design provisions. Very high and scattered SSR values can be observed for the shear strengths of WS critical specimens calculated using the EC-2 shear design provisions. In Fig. 8(a) and (b) it has been observed that the SSR values estimated using shear strength equations of IS 1343:2012, ACI 318-14, AASHTO LRFD and Cladera’s

strengths estimated using IS 1343:2012 and ACI 318-14 are more conservative at higher concrete strength as these shear design provisions are based on old test results with lower strength concrete. A uniform distribution of SSR values can be observed for both WS and FS failure specimens for SSR values calculated using UH and Cladera’s Methods. However, Cladera’s Method underestimates the shear strength of a significant number of WS as well as FS critical specimens with SSR values corresponding to dangerous categories of DPC. 5.3. Prestressing Force The prestressing force is used in evaluating the shear strength of PC beams by all shear strength estimation methods included in the present study except the UH Method. The effect of prestressing force has been studied with respect to the effective prestress in the prestressing tendon as well as the compressive prestress in the member cross-sections due to the applied prestress. The variation of SSRs with the compressive stress at the beam centroids due to the applied prestress is shown in Fig. 7. SSR values obtained from shear design provisions of IS 1343:2012 and ACI 318-14, which directly uses prestressing force to calculate the shear strength of PC members are plotted in Fig. 7(a). SSR values from EC-2, AASHTO LRFD and Cladera’s Method, which indirectly incorporate the prestressing force into the calculation of shear strength were plotted together in Fig. 7(b). A uniform distribution of SSR values can be observed in Fig. 7(c) for shear strength predictions using the UH Method. Extremely conservative and conservative predictions of shear strengths can be observed from IS 1343:2012 and ACI 318-14 code provisions respectively. It should be noted that for the ACI 318-14 method, the SSR values are uniformly distributed for flexure shear critical beams as shown in Fig. 7(a), even though the ACI 318-14 shear design equation does not

Fig. 7. Variation of SSR Values with Compressive Stress due to Prestress. 630

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Fig. 8. Variation of SSR Values with Effective Prestress.

category. AASHTO LRFD and EC-2 have higher SSR values at higher ρl for both web-shear and flexure-shear critical specimens, thereby indicating that the effect of ρl is not adequately implemented into these shear design methodologies. The same trend can also be observed in case SSR values obtained using IS 1343:2012 shear design equations, which considers the ρl term. However, the increment of SSR values obtained using IS 1343:2012 design equations is not as high as the SSR values obtained using AASHTO LRFD and EC-2 shear design equations. In contrast, the ACI 318-14 and UH Methods indicate that the implementation of ρl is not critical for prediction of shear strengths of prestressed concrete members with reasonable accuracy.

Method decrease with an increase in the effective prestress. This is more predominant for WS failure critical specimens than for FS critical specimens. However, UH Method (based on Loov’s concept) is unaffected by the effective prestress and has a uniform distribution of SSR values for the given range. Very high and scattered SSR values can be observed for the FS failed specimens in case of EC-2. Shear strengths predicted using IS 1343:2012 and AASHTO LRFD have very high SSR for webshear failure specimens. Based on the above observations, it can be concluded that the prestressing force is not a predominant variable for predicting the shear strength of PC members.

5.4. Longitudinal Reinforcement Ratio 5.5. Shear Reinforcement Index Cladera’s Method directly includes longitudinal reinforcement ratio (ρl) to calculate concrete contribution for both WS and FS critical beams. IS 1343:2012 uses ρl to calculate flexure-shear capacities of the beam. Variation of SSR values with ρl obtained using shear design provisions which directly use ρl has been plotted in Fig. 9(a). Variation of SSR values from shear design provisions like AASHTO LRFD and EC-2 which indirectly use ρl has been plotted in Fig. 9(b). Variation of SSR values from UH and ACI 318-14 shear strength equations which do not use ρl has been plotted together in Fig. 9(c). A uniform distribution of SSR values can be seen across the ρl values for the UH Method which does not consider the effect of ρl in its shear strength equations. The distribution of SSR values obtained using ACI 318-14 shear design equations, which also do not consider the ρl term, is more uniform for FS critical specimens than for WS critical specimens. There is a large scatter of SSR values at higher ρl values for shear strength predictions using the shear strength equations of IS 1343:2012 and Cladera’s Method even though these methods consider the ρl in their shear strength calculations. The SSR values obtained using Cladera’s Method for most of the specimens with low ρl fall under dangerous DPC

The shear strength of PC members significantly depends on the shear reinforcement index (ρvfy) of the members. Variation of SSR values obtained from IS 1343:2012, ACI 318-14 and UH Methods which are based on the 45-degree truss analogy has been plotted together against the ρvfy of the specimens in Fig. 10(a). Variation of SSR values obtained from EC-2 and Cladera’s Methods which use the variable angle truss analogy for steel contribution calculation has been plotted against the ρvfy of the specimens in Fig. 10(b). Variation of SSR values obtained from the AASHTO LRFD Method, which uses MCFT has been plotted separately against the ρvfy of the specimens in Fig. 10(c). It can be observed from Fig. 10 that the shear strength predictions from all the methods match better with the experimental results at higher values of ρvfy. This could be due to the higher steel contribution to the shear strength, thereby indicating that the ρvfy is an important parameter to estimate the shear strength of PC members. Shear strength prediction of code provisions is more conservative for web-shear critical specimens with lower shear reinforcement indices than flexure-shear critical specimens. Very high SSR values at lower values of ρvfy signify the absence

Fig. 9. Variation of SSR Values with Longitudinal Reinforcement Ratio. 631

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Fig. 10. Variation of SSR values with Shear Reinforcement Index.

1343:2012 maximum shear strength (Vmax) criteria significantly lower the actual shear strength contribution of concrete and steel, which leads to lower calculated shear strength values for beams over-reinforced in shear. This is clearly observed in the case of Ma’s [36] specimen (AR05908X) where the Vc + Vs equals to 1743 KN and Vexp is 2801 KN. However, IS 1343:2012 Vmax criteria limits Vcal to 801 KN. This causes a very high SSR of 3.50. Thus the conservative Vmax criteria in IS 1343:2012 results are very high SSR values.

or underestimation of concrete contribution to shear strength in EC-2, AASHTO LRFD and IS 1343:2012. It also confirms that the code provisions in EC-2 are not accurate for web-shear critical specimens, as suggested by Olalusi [63]. The SSR values of EC-2 and Cladera’s Method are similar for specimens with higher ρvfy since both methods have the same upper limits of shear strength governing such specimens. Shear strengths predicted using ACI 318-14 and IS 1343:2012 code provisions are conservative and extremely conservative respectively for specimens with higher ρvfy since the Vn, max equation (depending on the softened strength of concrete) governs at higher shear reinforcement index values. Shear strength of a large number of flexure-shear critical specimens with low ρvfy has been overestimated by Cladera’s Method. This indicates Cladera’s Method overestimates the concrete contribution to flexure-shear. A uniform distribution of SSR values has been observed for UH Method. However, the shear strength of some specimens with low shear reinforcement index has been over-estimated using the UH Method, as shown in Fig. 10(a).

6. Conclusions In this study, the accuracy and degree of the conservativeness of four shear design provisions and two shear strength equations for PC have been investigated. The conclusions drawn from this study are as follows.

• The developed comprehensive shear database of 274 PC beams

5.6. High Shear Strength Ratio Values SSR value of unity indicates the accurate prediction of shear strength by the shear design procedure or method. However, very high and low SSR values can be observed for some beams in Fig. 5 through Fig. 10. Very high SSR values have not been observed for shear strengths obtained using UH and Cladera’s Methods. However, Cladera’s Method overestimates the shear strength of the most number of specimens, which indicates the simplification used for shear strength calculation. Very high SSR values have been obtained for strength predictions using ACI 318-14, IS 1343:2012, EC-2, and AASHTO LRFD. Durrani [45] had investigated welded wire fabric as shear reinforcement for thin webbed prestressed concrete beams. For some of the specimens (Beam Nos.3 & 4) the shear reinforcement area (approximately 16.13 mm2) is close to the minimum shear reinforcement requirement [45]. EC-2 shear strength equation, which does not include the concrete contribution, results in very high SSR for these beams having lower shear reinforcement. Thus EC-2 shear design provisions result in very high SSR values when the shear reinforcement is nearer to the minimum requirement criteria. Sudhira et al. [33,34] had investigated PC beams with concentric prestressing force. High and very high SSR values have been observed for all these specimens, and the values are tabulated in Table1. EC-2 has very high SSR values because these specimens also have very low shear reinforcement. Thus concentric prestressing with lower shear reinforcement also resulted in very high SSR values in the case of EC-2. It is also observed that the restriction of the cotangent of the concrete strut angle to 2.5 also limited the steel contribution, resulting in very high SSR values for EC-2 code provisions. High SSR values for those beams were also obtained using AASHTO LRFD design equations indicating that the concrete contribution is significantly underestimated in these provisions. IS

• •

•

• 632

covers a different type of sections, grades of concrete, type of loading, amount of prestressing and stirrup steel and helps researchers to investigate new shear strength equations for both flexure-shear and web-shear critical specimens. A very few nontraditional shear failure tests have been reported on uniformly distributed loads and the shear database signifies the need to perform shear tests on PC beams under realistic load conditions with stirrups rather than three-point and four-point loading conditions. Shear strength predictions using the UH Method which is simple, matched more with the test results than the other shear design provisions and Cladera’s Method. The average SSR values from none of the code provisions and shear strength equations are less than unity. The average SSR value obtained using EC-2 shear design equations is the highest of the average SSR values obtained from all the code provisions and shear design equations considered in the present study. The average SSR values obtained from the UH Method and Cladera’s Method are the lowest. ACI 318-14 has the lowest SSR values among the shear code equations which signifies the need to modify the other shear design equations. Cladera’s Method overestimated the shear strength of most number of specimens. The upper limit of shear strength in Cladera’s Method is similar to the EC-2 maximum shear strength equation and governs the predicted shear strength of most of the test specimens. SSR values obtained from code provisions and Cladera’s Method are higher for large beams. Only the average SSR values of the large beams obtained using the UH Method is similar to the corresponding average SSR values of all the 274 beams in the developed database. Shear strengths of specimens with lower experimental shear strengths (Vexp) match well with the predicted strengths from the various shear strength equations. However, strength predictions for specimens with higher Vexp are less accurate from most of the shear

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Appendix A. Supplementary data

strength equations. Only shear strengths predicted by the UH Method matches well with lower as well as higher Vexp values, thereby resulting in the vertical distribution of histogram closer to unity with very few outliers in the box plot. The DPC score of shear code provisions is almost twice the DPC score of ACI 318-14 and shear strength equations. SSR values obtained from the shear code provisions and shear strength equations do not correspond to the extremely dangerous category. UH Method has the least DPC score and EC-2 method has the highest DPC score for large beams. DPC score of Cladera’s Method is close to the DPC score of the UH Method. However, the DPC score of Cladera’s Method for large beams is almost twice the corresponding DPC score of the UH Method. ACI 318-14 strength predictions matched more with test results for flexure-shear critical beams with a/d ratios greater than 2.5 indicating that the effect of a/d ratio also needs to be considered in the shear strength prediction for web-shear critical beams with a/d ratios less than 2.5. Cladera’s Method overestimates the shear strength of specimens in Kani’s valley and other methods like ACI 318-14, AASHTO LRFD, and EC-2 and IS 1343:2012 underestimates in the Kani’s valley. The SSR values obtained from methods which include √fc or fc2/3 directly or indirectly in the concrete contribution to shear strength are less scattered. Shear strengths predicted using EC-2, and AASHTO LRFD is more conservative at high concrete strength. Shear strengths calculated using ACI 318-14 and IS 1343:2012 shear design provisions are more conservative for higher strength specimens as these design provisions have been developed based on old test specimens with lower concrete strength. Predictions of shear strength matched more with experimental results for equations which did not include the effect of prestressing force, i.e., UH equation and ACI 318-14 flexure-shear strength equation. Other shear strength equations which included the effect of prestressing force (directly/indirectly) did not provide a uniform variation of SSR’s with the prestressing force thereby indicating that the prestressing force is not a predominant variable for predicting the shear strength of PC beams. SSR values decrease as the effective prestress increases and the predictions based on IS 1343:2012, ACI 318-14, AASHTO LRFD and Cladera’s Method overestimates the shear strength. A more uniform variation of SSR values with longitudinal steel ratios (ρ1) was obtained using ACI 318-14 for flexure-shear strength and UH Methods which do not include the effect of (ρ1) in their shear strength equations. Other design equations which include the effect of ρ1 resulted in non-uniform scattered variation of SSR values with ρ1, thereby indicating it is better to combine the effect of dowel action along with the arch action of the members in the shear strength equations. Shear strength of specimens predicted using the AASHTO LRFD design equations are less conservative at higher shear reinforcement index indicating that AASHTO LRFD provisions underestimate the steel contribution to shear strength. High SSR values at lower reinforcement ratio for EC-2 indicates the elimination of concrete contribution. Lower SSR values for Cladera’s Method confirms the overestimation of concrete contribution towards shear strength of PC specimens with stirrups. The upper limit of shear strengths governs the shear strengths prediction using the IS 1343:2012 design equations with the increase of shear reinforcement index value.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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